Mutual Inductance

PHYInductance & Transformers

Mutual induction is one coil inducing an EMF in a neighbouring coil because of their shared changing flux; the proportionality constant is the mutual inductance $M$ (henry). Students find the secondary EMF from $\varepsilon_2 = -M\,dI_1/dt$ , using either an instantaneous or an average rate of current change.

It is the principle behind the transformer.

Mutually induced EMF

\varepsilon_2 = -M\,\dfrac{dI_1}{dt}

M

= mutual inductance (H),

dI_1/dt

= rate of change of current in the primary (A/s);

\varepsilon_2

= EMF induced in the secondary (V).

Average EMF

|\varepsilon_2| = M\,\dfrac{\Delta I_1}{\Delta t}

\Delta I_1

= change in primary current,

\Delta t

= time interval; used when the current rises uniformly over a time.

Flux-linkage definition

M = \dfrac{N_2 \phi_{21}}{I_1}

N_2

= secondary turns,

\phi_{21}

= flux through one secondary turn due to the primary,

I_1

= primary current;

M

depends only on geometry/coupling.

$M$ depends on the coils' geometry, number of turns, separation and any core, NOT on the current or EMF.
Mutual inductance is symmetric: $M_{12} = M_{21} = M$ — the same constant works in either direction.
Convert milli-units: $5\,\text{mH} = 5\times10^{-3}\,\text{H}$ before substituting.
For a uniformly rising current use the AVERAGE rate $\Delta I/\Delta t$ ; the resulting EMF is the average EMF over that interval.
With a high-permeability core, $M$ increases roughly with $\mu_r$ , because the shared flux increases.
For two ideally coupled coils $M = \sqrt{L_1 L_2}$ ; in general $M = k\sqrt{L_1 L_2}$ with coupling coefficient $0 \le k \le 1$ .
The magnitude $|\varepsilon_2| = M\,|dI_1/dt|$ is reported positive; the minus sign only encodes opposition (Lenz).

Where the marks go

Forgetting to convert mH to H, giving an EMF $10^3$ too large.
Using the self-inductance EMF (with the secondary's own current) instead of the primary's $dI_1/dt$ for mutual induction.
Treating an average-rate problem as instantaneous or vice versa.
Assuming $M$ changes when the current changes — $M$ is a fixed geometric constant.

How the board asks it

Numericalthe mutually induced EMF formula
When the current in a primary coil changes uniformly from $2\,\text{A}$ to $10\,\text{A}$ in $0.1\,\text{s}$ , an average EMF of $4\,\text{V}$ is induced in a neighbouring coil. Calculate the mutual inductance $M$ of the pair of coils.
Define / statethe flux-linkage definition
Define the mutual inductance of a pair of coils and state its SI unit. Write the relation giving the EMF induced in the secondary coil in terms of $M$ and the rate of change of current in the primary.
Give reasonsfactors affecting mutual inductance
State the factors on which the mutual inductance of a pair of coils depends, and explain why $M$ does not depend on the current flowing through the primary coil.
Numerical $M = k\sqrt{L_1 L_2}$
Two coils of self-inductances $L_1 = 5\,\text{mH}$ and $L_2 = 20\,\text{mH}$ are wound so that they are perfectly coupled. Calculate the maximum possible mutual inductance between them.
Derive / provethe flux-linkage definition
Derive an expression for the mutual inductance of two long coaxial solenoids of length $l$ and area of cross-section $A$ , having $N_1$ and $N_2$ turns respectively.

Practise this topic

PreviousSelf-Inductance & Energy NextTransformer

Written for Sublevo. Question text quoted anywhere in these notes is the Council’s and carries its year and paper; the board’s own diagrams are not reproduced.